Question

The first angle of a triangle is one-third its second angle, whereas the third angle is 5 times the first angle. Find the measures of all the angles of the triangle.

Answer 1

Let, the three angles of the triangle be ${\theta }_1,{\theta }_2\text{and}{\theta }_3$

Given that,

The first angle is one-third of the second.

$\therefore {\theta }_1=\frac{{\theta }_2}{3}..........\left(i\right)$

Also, third angle is 5 times the first.

$\therefore {\theta }_3=5\times {\theta }_1$

Putting value of ${\theta }_1$ from $\left(i\right)$, we get:

$\Rightarrow {\theta }_3=5\times \frac{{\theta }_2}{3}...........\left(ii\right)$

We know that, the sum of all interior angles of a triangle is ${180}^{\circ }$

$\therefore {\theta }_1+{\theta }_2+{\theta }_3={180}^{\circ }$

Putting the values of ${\theta }_1\text{and}{\theta }_3$ from $\left(i\right)\text{and}\left(ii\right),$ we get:

$\Rightarrow \frac{{\theta }_2}{3}+{\theta }_2+\frac{5\times {\theta }_2}{3}={180}^{\circ }$

$\Rightarrow {\theta }_2={60}^{\circ }$

From $\left(i\right)$:

${\theta }_1={\frac{60}{3}}^{\circ }$

$\Rightarrow {\theta }_1={20}^{\circ }$

From $\left(ii\right)$:

${\theta }_3=5\times {\frac{60}{3}}^{\circ }$

$\Rightarrow {\theta }_3={100}^{\circ }$

Hence, the angle of triangles are ${20}^{\circ },{60}^{\circ }\text{and}{100}^{\circ }$.